What is the integral from a to 3a of x^2(x-a)^3 ?

The integral from a to 3a of x^2(x-a)^3 is (364a^6)/3-(726a^6)/5+51a^6+(a^6)/3

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integral from a to 3a of x^2(x-a)^3

\int_{a}^{3 a} x^{2} (x - a)^{3} d x

\frac{364 a ^{6}}{3} - \frac{726 a ^{6}}{5} + 51 a^{6} + \frac{a ^{6}}{3}

Solution steps

\int_{a}^{3 a} x^{2} (x - a)^{3} d x

Expand

x^{2} (x - a)^{3} : x^{5} - 3 a x^{4} + 3 a^{2} x^{3} - a^{3} x^{2}

= \int_{a}^{3 a} x^{5} - 3 a x^{4} + 3 a^{2} x^{3} - a^{3} x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{a}^{3 a} x^{5} d x - \int_{a}^{3 a} 3 a x^{4} d x + \int_{a}^{3 a} 3 a^{2} x^{3} d x - \int_{a}^{3 a} a^{3} x^{2} d x

\int_{a}^{3 a} x^{5} d x = \frac{364 a ^{6}}{3}

\int_{a}^{3 a} 3 a x^{4} d x = \frac{726 a ^{6}}{5}

\int_{a}^{3 a} 3 a^{2} x^{3} d x = 60 a^{6}

\int_{a}^{3 a} a^{3} x^{2} d x = a^{3} (9 a^{3} - \frac{a ^{3}}{3})

= \frac{364 a ^{6}}{3} - \frac{726 a ^{6}}{5} + 60 a^{6} - a^{3} (9 a^{3} - \frac{a ^{3}}{3})

Simplify

\frac{364 a ^{6}}{3} - \frac{726 a ^{6}}{5} + 60 a^{6} - a^{3} (9 a^{3} - \frac{a ^{3}}{3}) : \frac{364 a ^{6}}{3} - \frac{726 a ^{6}}{5} + 51 a^{6} + \frac{a ^{6}}{3}

= \frac{364 a ^{6}}{3} - \frac{726 a ^{6}}{5} + 51 a^{6} + \frac{a ^{6}}{3}

What is the integral from a to 3a of x^2(x-a)^3 ?
The integral from a to 3a of x^2(x-a)^3 is (364a^6)/3-(726a^6)/5+51a^6+(a^6)/3